arXiv subjects

Yoshitaka Okumura

Publications and source records attributed to Yoshitaka Okumura.

24 records · Page 2Linked to original sources

arXiv1996

BRST invariant Lagrangian of spontaneously broken gauge theories in noncommutative geometry

The quantization of spontaneously broken gauge theories in noncommutative geometry(NCG) has been sought for some time, because quantization is crucial for making the NCG approach a reliable and physically acceptable theory. Lee, Hwang and Ne'eman recently succeeded in realizing the BRST quantization of gauge theories in NCG in the matrix derivative approach proposed by Coquereaux et al. The present author has proposed a characteristic formulation to reconstruct a gauge theory in NCG on the discrete space $M_4\times Z_{_N}$. Since this formulation is a generalization of the differential geometry on the ordinary manifold to that on the discrete manifold, it is more familiar than other approaches. In this paper, we show that within our formulation we can obtain the BRST invariant Lagrangian in the same way as Lee, Hwang and Ne'eman and apply it to the SU(2)$\times$U(1) gauge theory.

hep-th↗

arXiv1995

New Incorporation of the Strong Interaction in NCG and Standard Model

The standard model is reconstructed by new method to incorporate strong interaction into our previous scheme based on the non-commutative geometry. The generation mixing is also taken into account. Our characteristic point is to take the fermion field so as to contain quarks and leptons all together which is almost equal to that of SO(10) grand unified theory(GUT). The space-time $M_4\times Z_2$; Minkowski space multiplied by two point discrete space is prepared to express the left-handed and right-handed fermion fields. The generalized gauge field $A(x,y)$ written in one-differential form extended on $M_4\times Z_2$ is well built to give the correct Dirac Lagrangian for fermion sector. The fermion field is a vector in 24-dimensional space and gauge and Higgs fields are written in $24\times24$ matrices. At the energy of the equal coupling constants for both sheets $y=\pm$ expected to be amount to the energy of GUT scale, we can get $\sin^2θ_{_{W}}=3/8$ and $m_{_{H}}=\sqrt{2}m_{_{W}}$. In general, the equation $m\ma{H}=(4/\sqrt {3})m\ma{W}\sinθ\ma{W}$ is followed. Then, it should be noticed that the same result as that of the grand unified theory such as SU(5) or SO(10) GUT is obtained without GUT but with the approach based on the non-commutative geometry and in addition the Higgs mass is related to other physical quantities as stated above.

hep-th↗

arXiv1995

Reconstruction of the spontaneously broken gauge theory in non-commutative geometry

The scheme previously proposed by the present authors is modified to incorporate the strong interaction by affording the direct product internal symmetry. We do not need to prepare the extra discrete space for the color gauge group responsible for the strong interaction to reconstruct the standard model and the left-right symmetric gauge model(LRSM). The approach based on non-commutative geometry leads us to presents many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete space; Minkowski space multipied by N-points discrete space. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needs N=2 discrete space for reconstruction in this formalism. \lr is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory(GUT). N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosons $ϕ$ and $ξ$ which are as usual transformed as (2,2*,0)$ and (1,3,-2) under left-handed SU(2)x right-handed SU(2)x U(1), respectively. xi is responsible to make the right handed-neutrino Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of phi.

hep-th↗

arXiv1995

Non-Commutative Differential Geometry and Standard Model

We incorporate Sogami's idea in the standard model into our previous formulation of non-commutative differential geometry by extending the action of the extra exterior derivative operator on spinors defined over the discrete space-time; four dimensinal Minkovski space multiplyed by two point discrete space. The extension consists in making it possible to require that the operator become nilpotent when acting on the spinors. It is shown that the generalized field strength leads to the most general, gauge-invariant Yang-Mills-Higgs Lagrangian even if the extra exterior derivative operator is not nilpotent, while the fermionic part remains intact. The proof is given for a single Higgs model. The method is applied to reformulate the standard model by putting left-handed fermion doublets on the upper sheet and right-handed fermion singlets on the lower sheet with generation mixing among quarks being taken into account. We also present a matrix calculus of the method without referring to the discrete space-time.

hep-th↗

arXiv1994

Gauge theory and Higgs mechanism based on differential geometry on discrete space M4 * ZN

Weinberg-Salam theory and $SU(5)$ grand unified theory are reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_{\mathop{}_{N}}$. Our starting point is the generalized gauge field expressed by $A(x,n)=\!\sum_{i}a^\dagger_{i}(x,n){\bf d}a_i(x,n), (n=1,2,\cdots N)$, where $a_i(x,n)$ is the square matrix valued function defined on $M_4\times Z_{\mathop{}_{N}}$ and ${\bf d}=d+\sum_{m=1}^{\mathop{}_{N}}d_{χ_m}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{χ_m}$ which is exterior derivative with respect to $Z_{\mathop{}_{N}}$ and the spontaneous breakdown of gauge symmetry is coded in ${d_{χ_m}}$. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Three sheets ($Z_3)$ are necessary for Weinberg-Salam theory including strong interaction and $SU(5)$ Gut. Our formalism is applicable to more realistic model like $SO(10)$ unification model.

hep-th↗

arXiv1994

Standard Model in Differential Geometry on Discrete Space M4*Z3

Standard model is reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_3$. $Z_3$ is necessary for the inclusion of strong interaction. Our starting point is the generalized gauge field expressed as $A(x,y)=\!\sum_{i}a^\dagger_{i}(x,y){\bf d}a_i(x,y), (y=0,\pm)$, where $a_i(x,y)$ is the square matrix valued function defined on $M_4\times Z_3$ and ${\bf d}=d+χ$ is generalized exterior derivative. We can construct the consistent algebra of $d_χ$ with the introduction of the symmetry breaking function $M(y)$ and the spontaneous breakdown of gauge symmetry is coded in ${d_χ}$. The gauge field $A_μ(x,y)$ and Higgs field $Φ(x,y)$ are written in terms of $a_i(x,y)$ and $M(y)$, which might suggest $a_i(x,y)$ to be more fundamental object. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Two model constructions are presented, which are distinguished in the particle assignment of Higgs field $Φ(x,y)$.

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